Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. \(a_{1}=5, \quad r=\frac{1}{5}\)

Step-by-Step Solution

Answer

The first five terms are 5, 1, \(\frac{1}{5}\), \(\frac{1}{25}\), \(\frac{1}{125}\).

Step 1: Understand the Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Here, the first term is given as \(a_1 = 5\) and the common ratio \(r = \frac{1}{5}\).

Step 2: Calculate the Second Term

To find the second term, multiply the first term by the common ratio. This means \(a_2 = 5 \times \frac{1}{5} = 1\).

Step 3: Calculate the Third Term

For the third term, multiply the second term by the common ratio. This gives \(a_3 = 1 \times \frac{1}{5} = \frac{1}{5}\).

Step 4: Calculate the Fourth Term

The fourth term is found by multiplying the third term by the common ratio: \(a_4 = \frac{1}{5} \times \frac{1}{5} = \frac{1}{25}\).

Step 5: Calculate the Fifth Term

Lastly, calculate the fifth term by multiplying the fourth term by the common ratio: \(a_5 = \frac{1}{25} \times \frac{1}{5} = \frac{1}{125}\).

Key Concepts

Common Ratio in Geometric Sequences

The common ratio is a fundamental characteristic of geometric sequences. It is the fixed number by which each term is multiplied to get the next term in the sequence. The common ratio is denoted by the letter \( r \). It can be a fraction, a decimal, or even a negative number. Understanding the common ratio helps in predicting the behavior of the sequence, such as whether the terms will grow or shrink.

• If \( |r| > 1 \), the sequence terms grow larger in magnitude.
• If \( |r| < 1 \), the sequence terms decrease and get closer to zero.
• If \( r = 1 \), all terms in the sequence are the same.

In our original exercise, the common ratio \( r \) is \( \frac{1}{5} \). This indicates the sequence's terms will shrink, as demonstrated by the decreasing values of the terms.

Understanding Sequence Terms

Sequence terms are the individual components of a sequence, labeled as \( a_1, a_2, a_3, \ldots \). Each term in a geometric sequence is calculated based on the previous term and the common ratio. Starting with the first term, which is often given, the rest can be derived.

• For \( a_1 = 5 \), the first term is already provided.
• Subsequent terms like \( a_2, a_3 \), etc., depend on the common ratio \( r = \frac{1}{5} \).

Determining these terms step by step helps you see the pattern and the effect of the common ratio on the size of the terms.The progression of values among sequence terms often allows students to predict future terms or solve backward to find earlier ones.

Multiplication in Sequences

Multiplication is the core operation in geometric sequences. Unlike arithmetic sequences that rely on addition, geometric sequences use multiplication to transition from one term to the next. This multiplicative process follows a simple rule: multiply the current term by the common ratio to find the next one.
In our example, we start with \( a_1 = 5 \). To find \( a_2 \), we multiply \( 5 \) by \( \frac{1}{5} \), resulting in \( a_2 = 1 \). This process continues:

• \( a_3 = 1 \times \frac{1}{5} = \frac{1}{5} \)
• \( a_4 = \frac{1}{5} \times \frac{1}{5} = \frac{1}{25} \)
• \( a_5 = \frac{1}{25} \times \frac{1}{5} = \frac{1}{125} \)

Multiplication consistently applies the common ratio to each term, demonstrating how the sequence evolves. Understanding this multiplication concept is key to manipulating and solving problems involving geometric sequences.